The quotient formulas on generalized Schur complements of positive semidefinite Hermitian matrices

The quotient properties on generalized Schur complements of positive semidefinite Hermitian matrices are studied with the properties of positive semidefinite Hermitian matrices and their generalized Schur complements. The quotient formulas on Schur complements are extended to generalized Schur complements of positive semidefinite Hermitian matrices by computing and comparing the block matrices. Based on this, the quotient formulas on some 1 generalized inverses involving generalized Schur complements of positive semidefinite Hermitian matrices are also obtained. The condition under which the quotient formula on Moore-Penrose generalized inverse satisfies is derived.